Questions: -frac172 cdotleft(-frac34+frac23right)^-2+left(-frac258right) cdotleft(-frac125+2right)-left(-frac130+frac25right)^-3 cdotleft(frac1130right)^2-frac14:left[-left(-frac12right)^2right]

Solution Steps

To solve the given expression, we need to break it down into smaller parts and evaluate each part step by step. Here's the high-level approach:

1. Evaluate the expression inside the first set of parentheses and then raise it to the power of -2.
2. Evaluate the expression inside the second set of parentheses and then multiply it by -25/8.
3. Evaluate the expression inside the third set of parentheses, raise it to the power of -3, and then multiply it by (11/30)^2.
4. Evaluate the expression inside the fourth set of parentheses, square it, and then divide -1/4 by this result.
5. Combine all the evaluated parts to get the final result.

Evaluate the expression inside the first set of parentheses and then raise it to the power of \(-2\): \[ -\frac{1}{72} \cdot \left(-\frac{3}{4} + \frac{2}{3}\right)^{-2} \] \[ -\frac{1}{72} \cdot \left(-\frac{1}{12}\right)^{-2} = -\frac{1}{72} \cdot 144 = -2 \]

Evaluate the expression inside the second set of parentheses and then multiply it by \(-\frac{25}{8}\): \[ \left(-\frac{25}{8}\right) \cdot \left(-\frac{12}{5} + 2\right) \] \[ \left(-\frac{25}{8}\right) \cdot \left(-\frac{2}{5}\right) = \frac{50}{40} = \frac{5}{4} \]

Evaluate the expression inside the third set of parentheses, raise it to the power of \(-3\), and then multiply it by \(\left(\frac{11}{30}\right)^{2}\): \[ \left(-\frac{1}{30} + \frac{2}{5}\right)^{-3} \cdot \left(\frac{11}{30}\right)^{2} \] \[ \left(\frac{11}{30}\right)^{-3} \cdot \left(\frac{11}{30}\right)^{2} = \frac{30}{11} \]

Evaluate the expression inside the fourth set of parentheses, square it, and then divide \(-\frac{1}{4}\) by this result: \[ -\frac{1}{4} : \left[-\left(-\frac{1}{2}\right)^{2}\right] \] \[ -\frac{1}{4} : \left[-\left(\frac{1}{4}\right)\right] = -\frac{1}{4} : -\frac{1}{4} = 1 \]

Final Answer